Applications of
Linear Programming
1
The Diet Problem
Find the cheapest combination of foods that will satisfy all your
nutritional requirements.
• Can be accessed interactively from the NEOS Server at:
http://www-neos.mcs.anl.gov/CaseStudies/dietpy/WebForms/
• For the results of an experiment with this, see the next
sheet.
• Practical considerations — what about taste, variety?
2
Portfolio Optimization
Minimize the risk in your investment portfolio subject to achieving a certain return. Can also be accessed from the NEOS
Server:
http://www.mcs.anl.gov/otc/Guide/CaseStudies/port
Many investment companies are now using optimization and
linear programming extensively to decide how to allocate assets.
The increase in the speed of computers has enabled the solution
of far larger problems, taking some of the guesswork out of the
allocation of assets.
1
2
3
Crew scheduling
ORD
- yJFK
: y
]
J
J
SFO J
y
J
J
CO
C
J
ATL
C
Jy
*
CX
yLAX
B
XXX
i
B
XXX
XXX IAH
B
X
z X
y
B
B
BN y
MIA
An airline has to assign crews to its flights.
• Make sure that each flight is covered.
• Meet regulations, eg, each pilot can only fly a certain amount
each day.
• Minimize costs, eg: accommodation for crews staying overnight
out of town, crews deadheading.
• Would like a robust schedule.
The airlines run on small profit margins, so saving a few percent
through good scheduling can make an enormous difference in
terms of profitability.
They also use linear programming for yield management.
3
4
Manufacturing and transportation
Japan
y
: H
HH
HH
H
Saudi Arabia HH
HH
y
HH
HH
H
H
j yLos Angeles
H
y Borneo
HH
1
A
H
HH
A
H
A
HH
A
H
HH A
H
jAU X
H
y
X
XX
z ?
y
Australia
New Zealand
For example, an oil company has oil fields in Saudi Arabia and
Borneo, refineries in Japan and Australia, and customers in the
US and New Zealand.
The fields produce different qualities of oil, which is refined
and combined into different grades of gasoline. What raw oil
should be shipped to which refinery? How much of each type
of gasoline should each refinery produce?
4
5
Telecommunications
Call routing: Many telephone calls from New York to Los
Angeles, from Houston to Atlanta, etc. How should these calls
be routed through the telephone network?
Network design: If we need to build extra capacity, which
links should we concentrate on? Should we build new switching
stations?
Internet traffic: For example, there was a great deal of construction of new networks for carrying internet traffic a few years
ago.
5
6
Traveling Salesman Problem
Given a set of cities, find the shortest route that visits each city
exactly once and returns to the home city.
Applications include:
• Vehicle routing — eg, how do we route school buses to
pick up children, how do Federal Express and UPS assign
packages to trucks?
• VLSI chip board manufacturing: holes need to be drilled
on the board. How do we route the drill to visit all these
holes so it takes the shortest possible route?
Website:
http://www.tsp.gatech.edu/
Largest problem solved to date has more than 24000 cities.
6
All 13509 cities in the continental USA with populations at least
500 in 1998, solved by Cook et al:
7